Problem 43 - Thomas Calculus

Question

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(2 n) !}$$

Step-by-Step Solution

Answer

The series converges by the Ratio Test, as the limit is \( \frac{1}{4} \lt 1 \).

Step 1: Understand the Series

The series to evaluate is given by \( \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!} \). Our task is to determine whether this series converges or diverges.

Step 2: Apply the Ratio Test

Apply the ratio test to determine convergence or divergence. For a series \( \sum a_n \), consider \( a_n = \frac{(n!)^2}{(2n)!} \). The ratio to evaluate is \( \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| \).

Step 3: Compute Ratio of Consecutive Terms

Determine \( \frac{a_{n+1}}{a_n} \) using:\[ a_{n+1} = \frac{((n+1)!)^2}{(2(n+1))!} = \frac{(n+1)^2(n!)^2}{(2n+2)(2n+1)(2n)!} \]Thus,\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^2 \cdot (2n)!}{(2n+2)(2n+1)(n!)^2} \]

Step 4: Simplify the Ratio

Further simplify:\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^2}{(2n+2)(2n+1)} \]\(\frac{(n+1)^2}{(2n+2)(2n+1)}\) simplifies to \( \frac{(n+1)^2}{4n^2 + 6n + 2} \).

Step 5: Evaluate the Limit

Compute the limit as \( n \to \infty \):\[ \lim_{n\to\infty} \frac{(n+1)^2}{4n^2 + 6n + 2} = \lim_{n\to\infty} \frac{n^2 + 2n + 1}{4n^2 + 6n + 2} \]Divide numerator and denominator by \( n^2 \):\[ \lim_{n\to\infty} \frac{1 + \frac{2}{n} + \frac{1}{n^2}}{4 + \frac{6}{n} + \frac{2}{n^2}} = \frac{1}{4} \]

Step 6: Conclude with the Ratio Test

Since the limit \( \frac{1}{4} \lt 1 \), by the Ratio Test, the series \( \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!} \) converges.

Key Concepts

Ratio Test

The ratio test is a useful tool when trying to determine if an infinite series converges or diverges. It is particularly handy when dealing with series that include factorials, like the series in our exercise. Here's how it works:

• Given a series \( \sum a_n \), observe the terms \( a_n \). In our exercise, \( a_n = \frac{(n!)^2}{(2n)!} \).
• Calculate the ratio \( \left|\frac{a_{n+1}}{a_n}\right| \), which is basically the next term divided by the current term.
• Take the limit of this ratio as \( n \) approaches infinity: \( \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| \).
• Evaluate this limit:
  • If the limit is less than 1, the series converges.
  • If the limit is greater than 1, the series diverges.
  • If the limit equals 1, the test is inconclusive.

In our example, solving this ratio eventually gave us \( \frac{1}{4} \), which is less than 1, indicating convergence.

Factorials in Series

Factorials, denoted by \( n! \), are products of all positive integers up to \( n \). They grow very quickly as \( n \) increases, often leading to large numbers in mathematical expressions. This rapid growth makes factorials both challenging and intriguing in analytical problems.

When dealing with series that contain factorials, like \( \sum \frac{(n!)^2}{(2n)!} \), understanding how they affect series is key:

• Factorials compare relative sizes of numerators and denominators, influencing convergence or divergence.
• In our exercise, the factorial in the denominator, \((2n)!\), grows significantly faster compared to the squared numerator \((n!)^2\).
• This discrepancy in growth ensures that terms of the series get smaller as \( n \) increases, leading towards convergence.

Factorials provide a structured way of tackling combinatorics, probability, and series problems, offering insights into the behavior of sequences and series.

Infinite Series

An infinite series is an expression that sums an infinite number of terms. These appear frequently in calculus and are useful in modeling and approximating complex mathematical functions. An infinite series can be written in the form \( \sum_{n=1}^{\infty} a_n \).

Understanding convergence or divergence of such series involves:

• Recognizing that convergent series approach a specific finite value as \( n \) increases.
• If a series diverges, it either does not approach any particular value or grows infinitely.
• Applying tests like the ratio test allows us to determine the behavior of infinite series efficiently.

In our exercise, we learned that applying the ratio test on the infinite series \( \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!} \) shows convergence, offering a deeper understanding of how such series behave over infinite sums. Infinite series are vital tools in mathematical analysis and have applications across various scientific fields.